The Open Operational Research Journal, 2008, 2, 1-7 1 1874-2432/08 2008 Bentham Science Publishers Ltd. Comparing Fundamentals of Additive and Multiplicative Aggregation in Ratio Scale Multi-Criteria Decision Making E.U. Choo * and W.C. Wedley Faculty of Business Administration, Simon Fraser University, B.C., Canada Abstract: Additive and multiplicative aggregations of ratio scale preferences are frequently used in multi-criteria decision making models. In this paper, we compare the advantages and limitations of these two aggregation rules by exploring only their fundamental properties after ratio scaled local priorities and criteria weights have been successfully generated from the decision maker. The comparisons of these properties are therefore independent of ancillary procedures such as interac- tive elicitations from decision makers, pairwise comparisons and calculations of local priorities and criteria weights. We compare six fundamental properties of the two aggregation rules. The criteria weights used in the multiplicative aggrega- tion have complicated meanings which are not well understood and often mixed up in the ambiguous notion of "criteria importance". As the scaling factors of the local preference values do not appear explicitly in the computations of the rela- tive ratios of the overall preferences in the multiplicative aggregation model, the relative ratios remain unchanged when the scaling factors are changed or an alternative is added or deleted. Furthermore, the relative ratios in the multiplicative aggregation do not depend on similar local preference values which cancel each other out mathematically. It is quite evi- dent that the additive aggregation model is superior and easier for decision makers to use and understand. We recommend the additive aggregation rule over the multiplicative aggregation rule. Keywords: Preferences, ratio scale, unit of measure, criteria weights, additive aggregation, multiplicative aggregation. 1. INTRODUCTION The basic MCDM problem in most decision making is to evaluate competing alternatives under multiple conflicting criteria. Even though rank ordering (an ordinal scale) of the alternatives is the most common form of solution sought by most decision makers (DM), it is always desirable to know the relative standings of the alternatives measured on a scale containing more information. Valuation on a ratio scale is preferred because it provides the DM with a relative measure of alternatives on each criterion as well as the overall ratio preference across all criteria. A ratio scale measure of overall preference value is also useful for the allocation of resources among all the alternatives. With a ratio scale, it is meaning- ful to reach conclusions such as "alternative A j is r times preferred to A k relative to all criteria". Additive aggregation and multiplicative aggregation of the local ratio preferences of each alternative into an overall preference are frequently used in multi-criteria decision making models [1-3]. In this paper, we compare the advantages and limitations of additive aggregation and multiplicative aggregation rules by exploring their fundamental properties. We do this with- out investigating the different approaches to ancillary and peripheral procedures such as interactive elicitations from decision makers, pairwise comparisons, and calculations of local priorities and criteria weights. By assuming that local priorities and criteria weights have been correctly derived, we are able to focus on the aggregation procedures. From the comparisons of the fundamental properties presented, it is quite evident that the additive aggregation rule is superior with simpler interpretations which are more readily under- stood by decision makers. *Address correspondence to this author at the Faculty of Business Admini- stration, Simon Fraser University, B.C., Canada; E-mail: choo@sfu.ca Fundamental basic elements of the MCDM framework are first depicted without any specific interpretations im- posed on these elements. It is assumed that no relevant crite- rion is missed and each criterion is autonomous. In section 3, the measures of criteria weight, local and overall preferences are assumed to be in ratio scale. Some necessary conditions and the role of normalization are discussed. We then give a brief literature review, with particular attention to the differ- ent ancillary procedures and contradicting opinions in model interpretations. Additive and multiplicative aggregation rules are formally introduced in Section 5. In Section 6, we elabo- rate and compare the fundamental properties of these aggre- gation rules. Finally, we summarize and give some conclu- sions. 2. BASIC ELEMENTS OF MCDM MODEL The basic elements of a typical MCDM model include a set A={A 1 ,A 2 ,…,A n } of n alternatives A 1 ,A 2 ,…,A n and a set C={C 1 ,C 2 ,…,C m } of m criteria C 1 ,C 2 ,…,C m . The effect of the criteria C 1 ,C 2 ,…,C m in C is represented by positive num- bers w 1 ,w 2 ,…,w m respectively. The vector w=[w 1 ,w 2 ,…,w m ] is called the criteria weight vector of the criteria C 1 ,C 2 ,…,C m in C. The criteria weight vector w is derived from question- ing the DM. The alternatives A 1 ,A 2 ,…,A n can be evaluated under each individual criterion C p , p=1,2,…,m. For each criterion C p (p=1,2,…,m), the local preference of the alterna- tives A 1 ,A 2 ,…,A n in A with respect to C p is represented by positive numbers x 1p ,x 2p ,…,x np , respectively. The vector x p =[x 1p ,x 2p ,…,x np ] is called the local preference vector of the alternatives A 1 ,A 2 ,…,A n in A with respect to C p . The local preference vectors x 1 ,x 2 ,…,x m are derived from questioning the DM. It is important to note that at this rudimentary level, we only assume that the numerical values in the criteria weight vector w and the local preference vectors x 1 ,x 2 ,…,x m exist